Stochastic Nature of Quanta-II | ||

22^{nd} October 2016 |
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"The
principle of Least Action is this: Of all the different sets of paths
along which a conservative system may be guided to move from one
configuration to another, with the sum of its potential and kinetic
energies equal to a given constant, that one for which the action is
the least is such that the system will require only to be started with
the proper velocities, to move along it unguided."
- Lord Kelvin and P. G. Tait, Treatise on Natural Philosophy, 1912. Note: The path of the least action is the one which is measured by Obs _{C} (v/c ~ 1), the observer with maximum capacity. This observer defines e-v system and 1_{AB} in our discussion. |
Continuing on, we shall discuss the relationship between the wave-function
and the discrete measurement space or j-space. We know that j-space
is a dynamic space, i.e. observers or objects can not be completely
still neither can they have infinite velocity. We need to find a
quantity which characterizes the path taken between two points A and B (problem of measurements). Let us call this quantity Action represented by S. Please note that the path AB exists on the unit-point sphere S
_{U}. Further we assume that the system is conservative, which means that the same path can be measured repeatedly without any loss of information. In j-space where every single measurement is unique, we can not have a truly conservative system unless we assume VT-symmetry, i.e. we must define the origin in the j-space first. We also have the information contained within the path AB, measurement of which will require certain amount of resources which are essentially the sum of Kinetic (T) and Potential (V) Energies. We can define a quantity H s.t. H = T+V. We call H the Hamiltonian of the path AB. We can correlate the Hamiltonian H and the Action S for the path AB, using Hamilton-Jacobi Equation: If action is independent of time i.e. ∂S/∂t = constant, then we can write the reduced form of Hamilton-Jacobi Equation as:
The reduced form of Hamilton-Jacobi Equation, is used by Schrödinger to derive the wave equation. ψ is
known as the wave-function. The time independence of Action S is
required so that H can be linearly solved in terms of q's. The action S
now can be written as
^{*}:
Ignoring the relativistic variation in mass, above equation is written as the familiar Schrödinger equation and the constant K is h/2π. We need to understand the equation S = K log _{e}ψ intuitively.Before we do that however, we must digress a little and discuss the concept of information, in terms of entropy first. Basically more the information, less the entropy (W) or the disorder (Ω) in the system. This is the reason, the Information is also known as negative-entropy or Negentropy. We can define, the Information corresponding to the path AB as measured by the macroscopic observer Obs _{M}, as:
The parameters Ω
represent number of measurements required to arrive at point B from
point A. For an observer who travels the path AB with one single
measurement (i.e. extracting all the available information with a
single measurement Ω _{0}), the entropy of the measurement-metric employed by this observer, will be zero since K'log_{e}(1) = 0. In our case Obs_{c} is the zero-entropy observer. Another observer Obs_{M} may make a very large number of measurements Ω_{M},
while traveling the same path AB. Obviously, the entropy of the
measurement-metric of the observer Obs_{M }will be very high. We can take the
difference in respective entropies and determine the information I_{AB} contained in path AB. This information can be translated in to the resources (~energy) needed by Obs_{M} to arrive at point B from A, while extracting all the information contained within the path AB.Let us now consider the equation S = K log _{e}ψ. Using the analogy above, let us write it for the path AB as follows:
S _{AB} - Action corresponding to the path AB.K _{AB}
- A constant with the unit of action, corresponding to the path AB. We
will discuss this very important constant in detail later.ψ _{M} - The measurements made by Obs_{M} while traveling the path AB. (This will be represented by a Gaussian or one of the stable probability distributions.)ψ _{0} - The measurements made by Obs_{c} while traveling the path AB. It is essentially a δ-function since Obs_{c} is a maximum capacity observer for the path AB. 1 _{AB} - The definition of 1 (~complete information content), for the path AB. The definition of
1_{AB}, will be based on the measurement made by Obs_{c}.We are interested in ψ _{M}. It represents a small fraction of sample measured by Obs_{M} from an infinitely large population, which is estimated by a probability distribution hence the stochastic nature of ψ_{M}. The
nature of the distribution would represent the type of problem being
estimated. These measurements are then compared with the zero-entropy
measurement. An illustrated representation is provided below:
_{U},
by an observer who within a limited time interval can measure only an
infinitesimally small number of samples out of a infinitely large
sample space. We note the value of Action is different for observers of
different capacities. For example the Action has a certain value for
the observer Obs_{c}
who can measure the information content of the path AB in single
measurement and this value is very low. At the same time for Obs_{M }measuring
the same information content, the action value will be very high, as the
observer capacity is fairly limited. The observer Obs_{M} will require lot more resources to move along the path AB unguided.Finally some remarks about K _{AB}, although we will discuss its significance in the information space later on. The coefficient K_{AB }is equal to the Boltzmann constant k_{B}
in lower information space i.e. thermodynamical system. But in the
discrete measurement space K_{AB} has much broader interpretation. It's
value is based on the measurement metric of the observer Obs_{C}. If we could calculate the total number of states in universe as measured by Obs_{M}, then this number will represent the number of measurements. Multiplying the number of states with the Planck constant h, will provide an estimate of resources required to measure the universe by Obs_{M}. Same measurement will be completed by Obs_{C} by using the resource value equal to h only. Furthermore since the Obs_{M} has much lower capacity, it can not measure value below h. Hence the value of Action S_{AB} for Obs_{C} will be an integral multiple of h ^{**}.For a macroscopic observer Obs _{M}, the constant K_{AB}
will represent different physical constants corresponding to Action
S, for different paths being measured i.e. different information
spaces and subsequently different nature of corresponding disorders Ω_{M}'s. The quantity Ω_{M }represents
macro-states in thermodynamic system and quantum-states for a quantum
system. For a lower information space (v/c << 1) K_{AB} will be the Boltzmann physical constant k_{B} (J.K^{-1}), whereas for higher information space (v/c ~ 1) K_{AB} will be the Planck universal constant h (J.s).Finally, does Quantum Mechanics describe the physical reality? The physical reality is based on the observer's measurements. QM provides a powerful method to estimate the physical reality as accurately as possible and it is reflected in the relation ∫| Ψ |_{M}^{2} dX = 1_{AB}.
Perhaps we should ask if there was more to the reality than the
physical reality as the observer Obs_{M }perceives it, based on the locality
condition? And if that was the case, how should we describe it?_______________________ * E. Schrödinger, "Quantization as a problem of proper values," Annalen der Physik (4), Vol. 79, 1926. An English translation is available under the title "Collected papers on Wave-Mechanics" published by Blackie and Son Limited. ** We have discussed this issue at a conceptual level earlier when we introduced ĥ, however we had not explained the correlation with K _{AB}. |
Previous Blogs:
Nutshell-2015 Chiral Symmetry Sigma-z and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous z-Axis Majorana ZFC Axioms Set Theory Nutshell-2014 Knots in j-Space Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, b-Field & Lower Mass Bound Incompleteness II The Supersymmetry The Cat in Box The Initial State and Symmetries Incompleteness I Discrete Measurement Space The Frog in Well Visual Complex Analysis The Einstein Theory of Relativity |

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